Fluctuations and mixing for planar random growth

Abstract

We study two models of random growth by aggregation on the plane, which turn out to share similar asymptotic features.

The first part of this thesis focuses on the Hastings-Levitov model HL(0), according to which clusters of particles are built on the complex plane by iterated composition of random conformal maps. Following the scaling limit result of Norris and Turner (2012), who proved that the limiting shape of large HL(0) clusters is a disc, we show that the fluctuations around this deterministic shape are described by a random holomorphic Gaussian field $\mathcal{F}$ on {|z| > 1}, of which we provide an explicit construction. We find that the boundary values of $\mathcal{F}$ perform an Ornstein-Uhlenbeck process on an infinite-dimensional Hilbert space, which can be characterised as the solution of a Stochastic Fractional Heat Equation. When the cluster is allowed to grow indefinitely, this boundary process converges to a log-correlated Gaussian Field, which coincides in law with the restriction of a Gaussian Free Field on the 2-dimensional torus to the unit circle {|z| = 1}.

The same scaling limit and boundary fluctuations are found by Jerison, Levine and Sheffield (2014) to arise in a different growth model, namely Internal Diffusion Limited Aggregation (IDLA). According to this discrete model, the aggregation process defines a Markov Chain on the infinite space of IDLA configurations, for which Jerison, Levine and Sheffield ask the following mixing question: how long does it take for IDLA dynamics to essentially forget where it started? We provide a partial answer to this question in the second part of this thesis, using coupling techniques to obtain an upper bound for this forget time. Finally, we specialise to IDLA on the cylinder graph $\mathbb{Z}$$_{N}$ x $\mathbb{Z}$, and show that our bound is polynomial in the size $\textit{N}$ of the base graph, as $\textit{N}$ $\rightarrow$ $\infty$.